# Transversality for local Morse homology with symmetries and applications

## TL;DR

This paper establishes transversality results for local Morse homology with symmetries, enabling applications in Hamiltonian dynamics, including local contact homology, persistence of iterations, and bifurcation analysis under symmetries.

## Contribution

It introduces new transversality techniques for symmetric Morse theory and applies them to Hamiltonian dynamics and symplectic topology.

## Key findings

- Proved a global existence theorem for symmetric Morse-Smale pairs.
- Established a local contact homology framework with persistence properties.
- Analyzed bifurcations of critical points under symmetries.

## Abstract

We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive perturbation process indexed by the strata of the isotropy set. A global existence theorem for symmetric Morse-Smale pairs is also proved. Regarding applications, we focus on Hamiltonian dynamics and rigorously establish a local contact homology package based on discrete action functionals. We prove a persistence theorem, analogous to the classical shifting lemma for geodesics, asserting that the iteration map is an isomorphism for good and admissible iterations. We also consider a Chas-Sullivan product on non-invariant local Morse homology, which plays the role of pair-of-pants product, and study its relationship to symplectically degenerate maxima. Finally, we explore how our invariants can be used to study bifurcation of critical points (and periodic points) under additional symmetries.